（理論研用）TeXコマンド集

• AFIR関数

$$\begin{equation}
F(\mathbf{Q})=E(\mathbf{Q})+\rho \alpha \frac{\displaystyle \sum_{i \in \mathrm{A}} \sum_{j \in B} \omega_{i j} r_{i j}}{\displaystyle \sum_{i \in A} \sum_{j \in B} \omega_{i j}}
\end{equation}$$

\begin{equation}
F(\mathbf{Q})=E(\mathbf{Q})+\rho \alpha \frac{\displaystyle \sum_{i \in \mathrm{A}} \sum_{j \in B} \omega_{i j} r_{i j}}{\displaystyle \sum_{i \in A} \sum_{j \in B} \omega_{i j}}
\end{equation}

• AFIR関数のパラメータω

$$\begin{equation}
\omega_{i j}=\left[\frac{\left(R_{i}+R_{j}\right)}{r_{i j}}\right]^{p}
\end{equation}$$

\begin{equation}
\omega_{i j}=\left[\frac{\left(R_{i}+R_{j}\right)}{r_{i j}}\right]^{p}
\end{equation}

• AFIR関数のパラメータα

$$\begin{equation}
\alpha=\frac{\gamma}{\left[2^{\frac{1}{6}}-\left(1+\sqrt{1+\dfrac{\gamma}{\epsilon}}\right)^{\frac{1}{6}}\right] R_{0}}
\end{equation}$$

\begin{equation}
\alpha=\frac{\gamma}{\left[2^{\frac{1}{6}}-\left(1+\sqrt{1+\dfrac{\gamma}{\epsilon}}\right)^{\frac{1}{6}}\right] R_{0}}
\end{equation}

• model-function

$$\begin{equation}
F(\mathbf{Q})=\frac{1}{2}\left(E^{\mathrm{X}}(\mathbf{Q})+E^{\mathrm{Y}}(\mathbf{Q})\right)+\frac{\left(E^{\mathrm{X}}(\mathbf{Q})-E^{\mathrm{Y}}(\mathbf{Q})\right)^{2}}{\alpha}
\end{equation}$$

\begin{equation}
F(\mathbf{Q})=\frac{1}{2}\left(E^{\mathrm{X}}(\mathbf{Q})+E^{\mathrm{Y}}(\mathbf{Q})\right)+\frac{\left(E^{\mathrm{X}}(\mathbf{Q})-E^{\mathrm{Y}}(\mathbf{Q})\right)^{2}}{\alpha}
\end{equation}

• DS-AFIR関数

$$\begin{equation}
F^{\mathrm{DS-AFIR}}(\mathbf{q})=E(\mathbf{q})+X\left\{Y|\mathbf{q}-\mathbf{p}|-(1-\mathrm{Y})\left|\mathbf{q}-\mathbf{q}_{0}\right|\right\} \
\end{equation}$$

\begin{equation}
F^{\mathrm{DS-AFIR}}(\mathbf{q})=E(\mathbf{q})+X\left\{Y|\mathbf{q}-\mathbf{p}|-(1-\mathrm{Y})\left|\mathbf{q}-\mathbf{q}_{0}\right|\right\}
\end{equation}

• 遷移状態理論に基づく速度定数

$$\begin{equation}
k=\frac{k_{B} T}{h} \exp \left[-\frac{\Delta \Delta G^{\ddagger}}{R T}\right]
\end{equation}$$

\begin{equation}
k=\frac{k_{B} T}{h} \exp \left[-\frac{\Delta \Delta G^{\ddagger}}{R T}\right]
\end{equation}

• Kohn-Sham 方程式

$$\begin{equation}
\left[-\nabla^{2}+V_{\text {ion }}(r)+V_{H}(r)+V_{x c}^{\sigma}(r)\right] \varphi_{i \sigma}(r)=\varepsilon_{i \sigma} \varphi_{i \sigma}(r)
\end{equation}$$

\begin{equation}
\left[-\nabla^{2}+V_{\text {ion }}(r)+V_{H}(r)+V_{x c}^{\sigma}(r)\right] \varphi_{i \sigma}(r)=\varepsilon_{i \sigma} \varphi_{i \sigma}(r)
\end{equation}

• ωB97X-D 汎関数

$$\begin{equation}
E_{xc}^{,\omega \mathrm{B} 97 \mathrm{X}}=E_{x}^{\mathrm{LR-HF}}+c_{x} E_{x}^{\mathrm{SR-HF}}+E_{x}^{\mathrm{SR-B} 97}+E_{c}^{\mathrm{B} 97}
\end{equation}$$

\begin{equation}
E_{xc}^{\,\omega \mathrm{B} 97 \mathrm{X}}=E_{x}^{\mathrm{LR-HF}}+c_{x} E_{x}^{\mathrm{SR-HF}}+E_{x}^{\mathrm{SR-B} 97}+E_{c}^{\mathrm{B} 97}
\end{equation}

• LR ハートリー・フォック交換項，SR ハートリー・フォック交換項

$$\begin{equation}
\begin{aligned}
E_{x}^{\mathrm{LR}-\mathrm{HF}}=&-\dfrac{1}{2} \sum_{\sigma} \sum_{i, j}^{\infty} \iint {\psi}_{i \sigma}^{\ast}\left(\mathbf{r}_{1}\right) {\psi}_{j \sigma}^{\ast}\left(\mathbf{r}_{1}\right) \\
& \times \frac{\operatorname{erf}\left(\omega r_{12}\right)}{r_{12}} {\psi}_{i \sigma}\left(\mathbf{r}_{2}\right) {\psi}_{j \sigma}\left(\mathbf{r}_{2}\right) d \mathbf{r}_{1} d \mathbf{r}_{2}
\end{aligned}
\end{equation}$$

$$\begin{equation}
\begin{aligned}
E_{x}^{\mathrm{SR}-\mathrm{HF}}=&-\dfrac{1}{2} \sum_{\sigma} \sum_{i, j}^{\infty} \iint {\psi}_{i \sigma}^{\ast}\left(\mathbf{r}_{1}\right) {\psi}_{j \sigma}^{\ast}\left(\mathbf{r}_{1}\right) \\
& \times \frac{\operatorname{erfc}\left(\omega r_{12}\right)}{r_{12}} {\psi}_{i \sigma}\left(\mathbf{r}_{2}\right) {\psi}_{j \sigma}\left(\mathbf{r}_{2}\right) d \mathbf{r}_{1} d \mathbf{r}_{2},
\end{aligned}
\end{equation}$$

\begin{equation}
\begin{aligned}
E_{x}^{\mathrm{LR}-\mathrm{HF}}=&-\dfrac{1}{2} \sum_{\sigma} \sum_{i, j}^{\infty} \iint \psi_{i \sigma}^{*}\left(\mathbf{r}_{1}\right) \psi_{j \sigma}^{*}\left(\mathbf{r}_{1}\right) \\
& \times \frac{\operatorname{erf}\left(\omega r_{12}\right)}{r_{12}} \psi_{i \sigma}\left(\mathbf{r}_{2}\right) \psi_{j \sigma}\left(\mathbf{r}_{2}\right) d \mathbf{r}_{1} d \mathbf{r}_{2},
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
E_{x}^{\mathrm{SR}-\mathrm{HF}}=&-\dfrac{1}{2} \sum_{\sigma} \sum_{i, j}^{\infty} \iint \psi_{i \sigma}^{*}\left(\mathbf{r}_{1}\right) \psi_{j \sigma}^{*}\left(\mathbf{r}_{1}\right) \\
& \times \frac{\operatorname{erfc}\left(\omega r_{12}\right)}{r_{12}} \psi_{i \sigma}\left(\mathbf{r}_{2}\right) \psi_{j \sigma}\left(\mathbf{r}_{2}\right) d \mathbf{r}_{1} d \mathbf{r}_{2},
\end{aligned}
\end{equation}